A Note on the Augmented Hessian When the Reduced Hessian is Semidefinite

Let H be a real symmetric n x n matrix, A an m x n real matrix, and Z a basis for the null space of A. When the reduced Hessian Z sup T HZ is positive definite, it is a standard result that there exists a finite p bar such that, for all p > p bar, the augmented Hessian H+pA sup T A is positive definite. In this note, we analyze the case when Z sup T HZ is positive semidefinite and singular, and show that the situation is more complicated. In particular, we give a simple necessary and sufficient condition for the existence of a finite p bar so that H+pA sup T A is positive semidefinite for p >= p bar. A corollary of our result is that if H is nonsingular and indefinite while Z sup T HZ is positive semidefinite and singular, then H+pA sup T A has a negative eigenvalue for any finite p.